Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities

Krupchyk, Katya; Uhlmann, Günther

doi:10.48550/arxiv.1909.08122

Cited by 27 publications

(38 citation statements)

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“…We emphasize that all functions on the right hand sides of (65) are independent of τ . Thanks to Proposition 2, see also (36) and (37), we also have b (j) k (0, 0) = b (j) 0,0 (0, 0) = 1, j = 1, 2.…”

Section: Step 4: Uniformity Of the Constant Cmentioning

confidence: 75%

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Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Lassas

Uhlmann

Wang

2018

Commun. Math. Phys.

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116

140

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We consider inverse problems in space-time (M, g), a 4-dimensional Lorentzian manifold. For semilinear wave equations g u + H(x, u) = f , where g denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map L : f → u|V , where V is a neighborhood of a time-like geodesic µ, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set where waves can propagate from µ and return back. Moreover, on a given space-time (M, g), the source-to-solution map determines some coefficients of the Taylor expansion of H in u .

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Section: Step 4: Uniformity Of the Constant Cmentioning

confidence: 75%

“…The research of inverse problems for non-linear equations is expanding fast. By using the higher-order linearization, inverse problems for nonlinear models have been studied for example in [3,11,12,13,16,17,21,20,32,36,37,38,41,45,46,53,60,62,63].…”

Section: Introductionmentioning

confidence: 99%

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Lassas

Uhlmann

Wang

2018

Commun. Math. Phys.

Self Cite

116

140

View full text Add to dashboard Cite

show abstract

“…Our main strategy is to use the linearization technique of Isakov and others [28,29,30,31,32] in dealing with nonlinear equations to decompose the inverse problem to the semilinear radiative transport equation ( 1) into an inverse coefficient problem for the linear transport equation where we reconstruct σ a and σ s by the result of Bal-Jollivet-Jungon [6], and an inverse source problem for the linear transport equation where we reconstruct the two-photon absorption coefficient σ b . This is the same type of strategy that have been successfully employed to solve many inverse problems for nonlinear PDEs recently; see for instance [4,11,13,14,15,18,20,25,26,33,35,36,38,39,40,41,43,44,45,46,47,50,58,61,62,63] and reference therein.…”

Section: Inverse Problems In the Radiative Transport Regimementioning

confidence: 99%

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Lai¹,

Ren²,

Zhou³

2021

Preprint

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Motivated by applications in imaging nonlinear optical absorption by photoacoustic tomography (PAT), we study in this work inverse coefficient problems for a semilinear radiative transport equation and its diffusion approximation with internal data that are functionals of the coefficients and the solutions to the equations. Based on the techniques of first-and secondorder linearization, we derive uniqueness and stability results for the inverse problems. For uncertainty quantification purpose, we also establish the stability of the reconstruction of the absorption coefficients with respect to the change in the scattering coefficient.

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“…We restrict ourselves to mentioning some of the important results known for quasilinear and semilinear elliptic and parabolic equations. For semilinear equations, see [8], [16], [18], [21], [23], [24], [27], [28], [29], [32], [31], [42]. For quasilinear equations, not in divergence form, see [22].…”

Section: Introductionmentioning

confidence: 99%

An inverse problem for the porous medium equation with partial data and a possibly singular absorption term

Cârstea¹,

Ghosh²,

Uhlmann³

2021

Preprint

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In this paper we prove uniqueness in the inverse boundary value problem for the three coefficient functions in the porous medium equation with an absorption term ǫ∂ t u−∇•(γ∇u m )+λu q = 0, with m > 1, m −1 < q < √ m, with the space dimension 2 or higher. This is a degenerate parabolic type quasilinear PDE which has been used as a model for phenomena in fields such as gas flow (through a porous medium), plasma physics, and population dynamics. We improve on previous work by obtaining uniqueness with a finite (rather than infinite) time of observation and also by introducing the additional absorption term λu q .

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Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities

Cited by 27 publications

References 0 publications

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

An inverse problem for the porous medium equation with partial data and a possibly singular absorption term

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